Last Time Electric field of a hollow sphere Electric field of a solid sphere + + E = 0 INSIDE THE SPHERE OUTSIDE THE SPHERE OUTSIDE THE SPHERE INSIDE THE SPHERE

Today Review of potential energy Electric potential Potential due to charges

Review: Single Particle Energy Energy of a Single Particle: Rest energy Kinetic Energy (v<<c) The Energy Principle for a Particle: W = Work done ON the particle If the rest energy does not change,

iClicker A horizontal force of 10 N pushes a bead along a wire. The wire has a length of 25 m. The horizontal displacement of the bead when it reaches the end of the wire is 10m. The vertical displacement is 1m. How much work was done moving the bead? Ignore gravity. 10 J 250 J 100 J 100 N L=25m dx dx F=10 N dx dx Dy = 1 m Dx=10 m

Review: Multiparticle Energy Principle Energy Principle for Each Particle: 1 2 Work done ON particle 1 Work done ON particle 2

Review: Multiparticle Energy Principle Energy Principle for Each Particle: 1 2 Work done ON particle 1 Work done ON particle 2 Total change in Single Particle Energies

Review: Potential Energy Total change in Single Particle Energies 1 2 just rearrange! Potential Energy is Meaningless for a Single Particle Change in Kinetic Energy + Change in Potential Energy of The System

Potential energy of charges Remember: potential energy comes from interaction of TWO objects We can find potential energy by checking the interaction of 2 particles q1 q2 Hold q1 fixed and move q2. How much work do we have to do?

Work to move q2 q1 q2 r a b Work we have to do against q1’s influence

Where did the Energy go? Assume vf = vi -- Then ΔK = 0. q1 q2 r a b Assume vf = vi -- Then ΔK = 0. Work always changes Esys, so the potential energy must have changed: Work done by the surroundings (our hand) ELECTRIC POTENTIAL ENERGY

iClicker Two particles with charge q sit a distance d apart. What is the potential energy of the system, including both particles? 2q1q2/4pe0d q1q2/4pe0d 2q1q2/4pe0d2 q1q2/4pe0d2 d q1 q2

What about circular motion? We’ve shown what the work is required to move 2 charges away or toward each other. What about moving 1 charge around each other?

Electric Potential Energy of Two Particles q2 q2 q1 q1 Uel > 0 for two like charges (repulsion) Uel < 0 for two opposite charges (attraction)

Electric and Gravitational Potential Energy q1 q2 m1 m2

Three Electric Charges Interaction between q1 and q2 is independent of q3 There are three interacting pairs: q1  q2 q2  q3 q3  q1 U12 U23 U31 U= U12+ U23+ U31

Multiple Electric Charges q1 q3 q6 Each (i,j) pair interacts: potential energy Uij q2 q5 q4

Electric Potential Electric potential  electric potential energy per unit charge Units: J/C = V (Volt) Volts per meter = Newtons per Coulomb Electric potential – often called potential Electric potential difference – often called voltage

V due to One Particle Single charge has no electric potential energy q2 Single charge has potential to interact with other charge – it creates electric potential probe charge J/C, or Volts

V due to Two Particles Electric potential is scalar: Electric potential energy of the system: q3 If we add one more charge q3:

Today Review of potential energy Electric potential Potential due to charges

Example The System = 2 charged plates + proton Uniform Electric Field QUESTION: As proton moves from A to B, what is the change in potential energy of The System? iClicker: The answer is... A) B) C) D) ANSWER: C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal). In a uniform field

Example The System = 2 charged plates + proton Uniform Electric Field QUESTION: As proton moves from A to B, what is the change in potential energy of The System? ANSWER: C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal).

Electric Potential With the "test charge" (proton) in the capacitor, there is potential energy between the proton and capacitor. Remove the "test charge" (proton)  E-field due to plates is still present ELECTRIC POTENTIAL is "the potential" to have potential energy if a test charge enters the system C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal).

Electric Potential (Uniform field) The System = 2 charged plates + proton Uniform Electric Field Test Charge C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal). This part exists independent of the test charge. It is "the potential" to have a potential energy difference This part is "The Potential Difference"

Electric Potential (Uniform field) The System = 2 charged plates + proton Uniform Electric Field V has units of "Volts" C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal). Units of Electric Field

Electric Potential (Uniform field) The System = 2 charged plates + proton Uniform Electric Field In this example, the change in Electric Potential is: C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal). Positive charges move toward lower voltages, like water running down a hill. Which is larger, VA or VB?

What's an eV? (Uniform field) The System = 2 charged plates + proton Uniform Electric Field An electron-Volt (eV) is the energy required to move q=1e through 1V. The proton lost of potential energy in this example. C The system is comprised of the charged capacitor plates and the electron. this example, the electric force is in the direction opposite to the displacement through which the force acts. Remember that e is a number. Also remember that the F in this expression is that due to sources within the system (F_internal).

Potential Difference: The Full Story Uniform E-Field, E||x: for uniform E||x For a uniform E-field pointing in any direction: If E is not uniform, but varies in space: POTENTIAL DIFFERENCE IN NONUNIFORM E-FIELD

Potential Difference: Path Independence Uniform E-Field: Uniform E-Field in a Capacitor

Potential Difference: Path Independence Uniform E-Field: Particle moves a distance d to the right. d Uniform E-Field in a Capacitor Part 1: Total is Independent of the Path taken a Part 2: d Part 3: y x

Potential Difference: Path Independence Path independence principle: V between two points does not depend on integration path

Today Electric Potential (Voltage relative to infinite separation) Potential Difference and Electric Field Path Independence of Potential Difference

Potential Difference and Electric Field dl F f For very short path: Example: E = 3.106 N/C, l = 1 mm:

Example: Different Paths near Point Charge 1. Along straight radial path: rf ri Origin at +q +q For final r greater than initial r, the change in potential is less than zero as expected since the path direction is the same as that of the electric field. Likewise, if we go from a larger r to a small r, the potential increases.

Example: Different Paths near Point Charge 2. Special case iA: AB: BC: + Cf:

Example: Different Paths near Point Charge 3. Arbitrary path +